The equation of the plane which passes through the line of intersection of planes r-n-1-q1- r-n-2-q2 and is parallel to the line of intersection of planes r-n-3-q3 and r-n-4-q4 is

The correct option is A [n⃗_2 n⃗_3 n⃗_4](r⃗·n⃗_1 q_1)=[n⃗_1 n⃗_3 n⃗_4](r⃗·n⃗_2 q_2) The equation of the plane which passes through the line of intersectio ...

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Byju's Answer

Standard XII

Mathematics

Equation of a Plane : Intercept Form

The equation ...

Question

The equation of the plane which passes through the line of intersection of planes $\to r \cdot {\to n}_{1} = q_{1}, \to r \cdot {\to n}_{2} = q_{2}$ and is parallel to the line of intersection of planes $\to r \cdot {\to n}_{3} = q_{3}$ and $\to r \cdot {\to n}_{4} = q_{4}$ is

[{\to n}_{2} {\to n}_{3} {\to n}_{4}] (\to r \cdot {\to n}_{1} - q_{1}) = [{\to n}_{1} {\to n}_{3} {\to n}_{4}] (\to r \cdot {\to n}_{2} - q_{2})

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[{\to n}_{1} {\to n}_{2} {\to n}_{3}] (\to r \cdot {\to n}_{4} - q_{4}) = [{\to n}_{4} {\to n}_{3} {\to n}_{1}] (\to r \cdot {\to n}_{2} - q_{2})

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[{\to n}_{4} {\to n}_{2} {\to n}_{1}] (\to r \cdot {\to n}_{4} - q_{4}) = [{\to n}_{1} {\to n}_{2} {\to n}_{3}] (\to r \cdot {\to n}_{2} - q_{2})

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Solution

The correct option is A $[{\to n}_{2} {\to n}_{3} {\to n}_{4}] (\to r \cdot {\to n}_{1} - q_{1}) = [{\to n}_{1} {\to n}_{3} {\to n}_{4}] (\to r \cdot {\to n}_{2} - q_{2})$
The equation of the plane which passes through the line of intersection of planes $\to r \cdot {\to n}_{1} = q_{1}$ and $\to r \cdot {\to n}_{2} = q_{2}$ is
$P : \to r \cdot {\to n}_{1} - q_{1} + k (\to r \cdot {\to n}_{2} - q_{2}) = 0$
Where $k$ is real
$\Rightarrow \to r \cdot ({\to n}_{1} + k \cdot {\to n}_{2}) - q_{1} - k q_{2} = 0$
Since, $P$ is parallel to the line of intersection of planes,
$\to r \cdot {\to n}_{3} = q_{3}$ and $\to r \cdot {\to n}_{4} = q_{4}$
Therefore, $({\to n}_{1} + k \cdot {\to n}_{2}) . ({\to n}_{3} \times {\to n}_{4}) = 0$

$\Rightarrow k = - \frac{{\to n}_{1} . ({\to n}_{3} \times {\to n}_{4})}{{\to n}_{2} . ({\to n}_{3} \times {\to n}_{4})} = - \frac{[{\to n}_{1} {\to n}_{3} {\to n}_{4}]}{[{\to n}_{2} {\to n}_{3} {\to n}_{4}]}$
Substituting $k$ in $P$ , we get
$[{\to n}_{2} {\to n}_{3} {\to n}_{4}] (\to r \cdot {\to n}_{1} - q_{1}) = [{\to n}_{1} {\to n}_{3} {\to n}_{4}] (\to r \cdot {\to n}_{2} - q_{2})$
Ans: A

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The equation of the plane which passes through the line of intersection of planes →r⋅→n1=q1,→r⋅→n2=q2 and is parallel to the line of intersection of planes →r⋅→n3=q3 and →r⋅→n4=q4 is

The equation of the plane which passes through the line of intersection of planes $\to r \cdot {\to n}_{1} = q_{1}, \to r \cdot {\to n}_{2} = q_{2}$ and is parallel to the line of intersection of planes $\to r \cdot {\to n}_{3} = q_{3}$ and $\to r \cdot {\to n}_{4} = q_{4}$ is